LETTERS
Accurate theoretical fits to laser-excited
photoemission spectra in the normal phase
of high-temperature superconductors
PHILIP A. CASEY1 , J. D. KORALEK2,3 , N. C. PLUMB2 , D. S. DESSAU2,3 AND PHILIP W. ANDERSON1 *
1
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
3
JILA, University of Colorado and NIST, Boulder, Colorado 80309, USA
* e-mail: pwa@princeton.edu
2
Published online: 13 January 2008; doi:10.1038/nphys833
1.50
1.45
1.40
γ (x )
It has long been believed that the ‘normal’ (nonsuperconducting) state of the high-transition-temperature
superconductors is anything but normal1–3 . In particular,
this state, which exists above the superconducting transition
temperature T c , has very unusual transport properties4,5 and
electron spectral functions6,7 , presenting a more difficult,
complex and important problem than the superconductivity
itself. The origin of this difficulty and complexity resides
in the strong electronic correlations, or the many-body
Coulomb interactions between electrons, which cannot be
properly treated within the standard theories of the electronic
structure of solids. A new treatment of these interactions,
on the basis of a Gutzwiller projection—which gives zero
quasiparticle weight at the Fermi surface and removes the
possibility for double electron occupancy on any one site—
has recently been proposed8 , but fits to available data were
unsatisfactory. Here, we compare the electron spectral functions
computed within this theoretical treatment with bulk-sensitive
measurements made by low-energy photons, using laserexcited angle-resolved photoemission spectroscopy of the
superconductor Bi2 Sr2 CaCu2 O8+δ (refs 9,10). The theory captures
the asymmetrical shape of the experimental curves with good
accuracy and in principle has only one free parameter. Moreover,
no background subtraction is necessary.
The most direct way to measure the electronic excitation
spectrum of a solid is angle-resolved photoemission spectroscopy
(ARPES), which gives the energy- and momentum-dependent
spectral function11 . Historically, results of these measurements on
cuprate superconductors have given broad spectral features as well
as large backgrounds6,7 . These features were not well understood,
but generically have been described as representative of short-lived
electronic states or totally incoherent electronic states such as would
be the case in a system with spin–charge separation. Recently
however, much sharper ‘quasiparticle-like’ electronic excitations
have been observed using the laser-excited ARPES technology
developed in Colorado9,10 . The low photon energy in these
experiments (6 eV as compared with the typical 20–50 eV) greatly
increases both the momentum and energy resolution, reduces
the final-state broadening effects, decreases the background and
increases the bulk sensitivity, each of which overcomes concerns
of the previous experiments9,10 . Thus, it is believed that the new
1.35
1.30
1.25
1.20
0.10
0.15
0.20
0.25
x
Figure 1 Infrared spectrum exponents for Bi2 Sr2 CaCu2 O8+δ . Data points from
ref. 13 with linear best fit of ref. 13 (dashed line) and predicted value from ref. 8
(solid line). The predicted exponent stems from σ (ω) = (iω) −2+γ with γ = 1 + 2p,
and p is given in the text.
laser-excited ARPES spectra represent the most accurate cuprate
spectral line shapes so far. A theoretical understanding of these
spectra is clearly required.
One of the most difficult aspects of developing a theory of the
electronic structure of the cuprate superconductors is dealing with
the coulombic interactions between the doped holes (absence of
electrons, which are the carriers of electric current). In particular,
a double occupancy of holes on a single lattice site is energetically
unfavourable compared with single or zero occupancy, a fact that
breaks the electron–hole symmetry about the half-filled case. In
1963, Gutzwiller presented his projection operator to deal with
the Coulomb interactions, by which the probability amplitude for
double occupancy is set to zero12 . A formalism to deal with the
Gutzwiller projection for the cuprates has recently been developed
by one of us8 , and was found to have the unusual property that
there are no true quasiparticle excitations, that is, the quasiparticle
residue at the Fermi level is zero. Despite this, here we show that
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LETTERS
200 K
vFk = –29.7 meV
Γ = 17.2 meV
B L = 0.10
150 K
vFk = –2.82 meV
Γ = 10.1 meV
B L = 0.096
Intensity (arb. units)
Intensity (arb. units)
c
b
200 K
vFk = –1.75 meV
Γ = 15.6 meV
B L = 0.15
–100
Energy (meV)
150 K
vFk = –29.5 meV
Γ = 13.8 meV
B L = 0.088
–200
0
150 K
vFk = –49.8 meV
Γ = 20.1 meV
B L = 0.14
100 K
vFk = –51.2 meV
Γ = 17.0 meV
B L = 0.12
100 K
vFk = –28.7 meV
Γ = 10.5 meV
B L = 0.060
100 K
vFk = –2.65 meV
Γ = 7.34 meV
B L = 0.080
–200
200 K
vFk = –48.3 meV
Γ = 23.5 meV
B L = 0.14
Intensity (arb. units)
a
–100
Energy (meV)
0
–200
–100
Energy (meV)
0
Figure 2 Laser-excited ARPES EDCs in the strange-metal phase above Tc of optimally doped Bi2 Sr2 CaCu2 O8+δ . a–c, k near k F (a) and at higher quasiparticle energies
(b,c), as quoted. Data points are experimental and dashed curves are fitted lorentzians with background, B L f(ω/T ). Solid curves are theoretical fits from the present paper,
equation (1). Backgrounds are measured in units of the intensity relative to the peak of the EDC.
this quasiparticle-less theory produces excellent fits to the sharp
laser-excited ARPES spectra, which look more quasiparticle-like
than any previous ARPES spectra.
In ref. 8, it was shown that the effect of the Gutzwiller projection
is to multiply the free-particle Green’s function in space-time by
a factor t −p , where p is (1/4)(1 − x)2 and x is the hole doping
level. This value of the exponent is approximately confirmed by the
exponent of the infrared conductivity dependence on frequency13 ,
as shown in Fig. 1. Motion of a particle near the Fermi surface
is essentially one-dimensional, so we may take the free-particle
Green’s function in space-time as 1/(x − vF t ). To get the imaginary
part of G (the density) in k and frequency space, we must Fourier
transform G(x, t ),
ZZ
G(k, ω) =
dx dt ei(kx−ωt ) t −p /(x − vF t ).
Doing the x integration by a contour integration (the sign of t
determines which way to close the contour), this becomes,
Z
G(k, ω) = dt t −p ei(vF k−ω)t ∝ (vF k − ω)−1+p .
The imaginary part of this expression is the T = 0 energy
distribution curve (EDC). If p = 0, this is just a delta function at
the quasiparticle energy, vF k, but if p is finite it has an imaginary
part for all ω > vF k. The quasiparticle becomes a cut singularity,
not a pole, in the complex plane and does not have a finite residue
at the singularity, that is, it has quasiparticle residue Z = 0.
The absence of a finite Z has a profound effect on the
temperature behaviour. If there are ordinary quasiparticles, their
energies are not affected by thermal fluctuations. Impurity
scattering, for instance, only changes their spatial wavefunctions—
and there is no tendency towards ‘ω, T scaling’, which we naively
expect from the analogy between the Boltzmann distribution
e−E/kB T and the Schrodinger time dependence eiH t . In other words,
in that case there are conservation laws, effected by Ward identities,
which restrict the scattering, whereas in the current case there are
only phase space restrictions, so that the effect of raising the energy
is the same as raising the temperature. A heuristic approximation
to the effect of finite T is to insert a relaxation rate Γ = AT , where
A is a constant of order unity, so that we replace t −p by t −p e−Γ t .
Yuval has suggested that for finite T , t → sinh(πT t )/πT (ref. 14),
so that we might expect A ∼ πp, but that is only an estimate. The
excitations into which any decay takes place are fermionic, so that
we must also multiply by the Fermi function of energy hω/2π.
The observed spectra clearly show a broadening that increases
with increasing binding energy. A conventional Fermi liquid would
have the scattering proportional to ω2 , where ω is the difference
from the Fermi energy, producing further decay for the curves with
k farther from kF . In a supplement for ref. 8, it is shown that there
is an underlying, ‘hidden’ Fermi liquid of excitations, which can
therefore be expected to be scattered at the conventional rate ∼ω2
(ref. 15). The final expression for the EDC then becomes,
f (ω/T )
Intensity = Im{G} = Im
[(vF k − ω) + iΓ ]1−p
= f (ω/T )
sin[(1 − p)(π/2 − tan−1 [(ω − vF k)/Γ ])]
.
[(ω − vF k)2 + Γ 2 ](1−p)/2
(1)
Here, we have expressed all frequencies and temperatures in the
same energy units, and f is the Fermi distribution,
1
.
f (ω/T ) = h̄ω/k T
e B +1
The line shape, equation (1), is the Doniach–Sunjic line shape16
multiplied by the Fermi function. Although it shares with the
simple power law the fact that the singularity is not a pole and has
Z = 0, it was historically often mistaken for a simple lorentzian.
In principle, it is possible to calculate the temperature
dependence of the Green’s function, and its Fourier transform
G(k, ω), in a less heuristic way, because Yuval has given a
prescription for the Green’s function in the X-ray problem at finite
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211
LETTERS
30
25
Γ (meV)
20
200 K
15
150 K
10
100 K
A 200 K = 0.87 C 200 K = 3.5 eV –1
5
A 150 K = 0.81 C 150 K = 3.6 eV –1
A 100 K = 0.88 C 100 K = 3.6 eV –1
0
–60
–50
–40
–30
v F (k – k F) (meV)
–20
–10
0
Figure 3 Relaxation rate, Γ , (data points) extracted by fitting the experimental
EDCs. Fits were equation (1). Solid lines are best fits of Γ = Ak B T + Cv F2 (k − k F ) 2 .
The given empirical values fit well to universal parameters A = 0.85 and
C = 3.6 eV−1 .
T (ref. 14). However, the formalism would involve enormously
complicated contour integration and we think the above heuristic
form is adequate for the current purposes. In explicit form,
equation (1) is quite complicated, and it should be: the central
portion resembles a lorentzian, whereas the tail behaves like ω−1+p .
Figure 2 compares the theoretical predictions of equation (1)
with the measured laser-excited ARPES EDCs at the three k
points and three temperatures from ref. 9 (many more k points
were fitted as seen in Fig. 3). Lorentzian fits to the data are also
shown, similar to those given in ref. 9. The fits using equation (1)
did not include any background term, whereas a small energyindependent background was added for the lorentzian fits, as
this was necessary for them to obtain reasonable fidelity. A value
p = 0.18, corresponding to the optimal hole doping level x = 0.15
was used. The sample was nearly optimally doped Bi2 Sr2 CaCu2 O8+δ
with a transition temperature of around 90 K, and k was along the
nodal direction, so at the temperatures between 100 and 200 K there
is little or no effect of the superconducting gap. The parameters for
each curve are also given in Fig. 2.
The fits using equation (1) are superior to the pure lorentzian
fits at capturing the asymmetry of the measured EDCs, especially
for the spectra far from the Fermi energy, EF . In principle, these new
fits have at most one truly free parameter—an approximation leaves
one numerical coefficient free within narrow limits. The current
fits are not perfect, nor should they be—the exponential cutoff
assumed is probably too slow, and we might expect a steeper falloff
on the low-energy side. We also note that no extra background term
is needed to fit with the strange-metal line shape, whereas one is
needed to fit the pure lorentzians (this background amplitude is
reported in Fig. 2 as BL ). We have found that the spectral functions
of Fermi liquid and marginal Fermi liquid theories can also capture
the asymmetry of the data better than pure lorentzians. However,
these theories were not found to fit the data set across multiple
temperatures with a universal set of parameters as the strange-metal
theory does, and therefore a detailed discussion of them will be
reserved for a future publication by J.D.K. et al.
Figure 3 shows the trends of the extracted relaxation rate, Γ ,
as a function of temperature and binding energy. A similar plot
for the lorentzian fits can be seen in Fig. 3b of ref. 9. The current
fits show a slightly smaller Γ and a smoother behaviour, which we
parameterize as Γ = AkB T + CvF2 (k − kF )2 . As seen in Fig. 3, the
extracted A and C values are nearly constant over all the individual
EDC fits, implying a near-universal set of fits.
The spectral line shapes predicted by the strange-metal theory
of cuprate superconductors are found to fit the laser-excited ARPES
EDCs accurately and with a minimal number of free parameters.
Compared with simple lorentzian fits, the asymmetry in the EDCs
is well characterized and no background term is required. It should
now be possible, with considerably more computational effort, to
analyse the curves in the superconducting state and understand the
subtle intensity transfers that occur there.
Received 18 July 2007; accepted 6 December 2007; published 13 January 2008.
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Acknowledgements
P.A.C. acknowledges support from a National Sciences and Engineering Research Council of Canada
Postgraduate Scholarship. Support for J.D.K., N.C.P. and D.S.D. came from DOE Grant No.
DE-FG0203ER46066, with other funding from NSF Grant No. DMR 0706657, and by the NSF ERC for
Extreme Ultraviolet Science and Technology under NSF Award No. 0310717. J.D.K., N.C.P. and D.S.D.
thank J. F. Douglas, Z. Sun, A. Fedorov, J. Griffith, S. Cundiff, M. Murnane and H. Kapteyn for help
with the experiments and Y. Aiura, H. Eisaki and K. Oka for growing the Bi2212 samples.
Correspondence and requests for materials should be addressed to P.W.A.
Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/
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